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Relative Equilibria in the 3-Dimensional Curved \(n\)-Body Problem

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Florin Diacu

The author considers the \(3\)-dimensional
gravitational \(n\)-body problem, \(n\ge 2\), in spaces
of constant Gaussian curvature \(\kappa\ne 0\), i.e. on
spheres \({\mathbb S}_\kappa^3\), for \(\kappa>0\), and
on hyperbolic manifolds \({\mathbb H}_\kappa^3\), for
\(\kappa<0\). His goal is to define and study relative
equilibria, which are orbits whose mutual distances remain constant in
time. He also briefly discusses the issue of singularities in order to
avoid impossible configurations. He derives the equations of motion
and defines six classes of relative equilibria, which follow naturally
from the geometric properties of \({\mathbb S}_\kappa^3\) and
\({\mathbb H}_\kappa^3\). Then he proves several criteria, each
expressing the conditions for the existence of a certain class of
relative equilibria, some of which have a simple rotation, whereas
others perform a double rotation, and he describes their qualitative
behaviour.

Title (HTML):
Relative Equilibria in the 3-Dimensional Curved \(n\)-Body Problem

Author(s) (Product display):
Florin Diacu

Affiliation(s) (HTML):
University of Victoria, Victoria, B.C., Canada

Abstract:

The author considers the \(3\)-dimensional
gravitational \(n\)-body problem, \(n\ge 2\), in spaces
of constant Gaussian curvature \(\kappa\ne 0\), i.e. on
spheres \({\mathbb S}_\kappa^3\), for \(\kappa>0\), and
on hyperbolic manifolds \({\mathbb H}_\kappa^3\), for
\(\kappa<0\). His goal is to define and study relative
equilibria, which are orbits whose mutual distances remain constant in
time. He also briefly discusses the issue of singularities in order to
avoid impossible configurations. He derives the equations of motion
and defines six classes of relative equilibria, which follow naturally
from the geometric properties of \({\mathbb S}_\kappa^3\) and
\({\mathbb H}_\kappa^3\). Then he proves several criteria, each
expressing the conditions for the existence of a certain class of
relative equilibria, some of which have a simple rotation, whereas
others perform a double rotation, and he describes their qualitative
behaviour.